A Large-Scale Stochastic Gradient Descent Algorithm Over a Graphon

We study the large-scale stochastic gradient descent algorithm over a graphon with a continuum of nodes, which is regarded as the limit of the distributed networked optimization as the number of nodes goes to infinity. Each node has a private local cost function. The global cost function, which all nodes cooperatively minimize, is the integral of the local cost functions on the node set. We propose a stochastic gradient descent algorithm evolving as a graphon particle system, where each node heterogeneously interacts with others through a coupled mean field term. It is proved that if the graphon is connected, then by properly choosing the algorithm gains, all nodes' states achieve consensus uniformly in mean square. Furthermore, if the local cost functions are strongly convex, then all nodes' states converge uniformly to the minimizer of the global cost function in mean square.